Finite Group Databasebeta

← #14480689

Subgroups — #14480689

Order 1920.

3112 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6782C212yesmaximal, complemented
2#11756252C211yesmaximal, characteristic, complemented
3C5×#5493C211yesmaximal, characteristic, complemented
4C2×C3×#4023C211yesmaximal, characteristic, complemented
5#11755428C212yesmaximal, complemented
6#11754331C214yesmaximal, complemented
7#11754330C214yesmaximal, complemented
8#11752874C214yesmaximal, complemented
9#11752873C214yesmaximal, complemented
10#11752872C214yesmaximal, complemented
11#11752871C214yesmaximal, complemented
12#10618195—31nomaximal, Hall
13C22×#1220C2212yes—
14C22×C5×#67C2212yes—
15C23×C3×Dic5C2211yescharacteristic
16C22×#1204C2212yescomplemented
17C22×#1203C2212yescomplemented
18C22×#1200C2212yescomplemented
19C2×#6782—216no—
20C2×#6782C2214yes—
21C2×#6782C2214yes—
22C2×#6782C2214yescomplemented
23C2×#6782C2214yescomplemented
24C22×D3×Dic5C2212yescomplemented
25C2×C3×C5×C22⋊C4C2211yescharacteristic
26#91819C2214yescomplemented
27#91775C2214yescomplemented
28#91771C2214yescomplemented
29C5×#930C2214yescomplemented
30C5×#886C2214yescomplemented
31C5×#882C2214yescomplemented
32C2×C3×C22⋊Dic5C2211yescharacteristic
33C2×C3×#683C2212yescomplemented
34C2×C3×C4×Dic5C2212yescomplemented
35C3×#4023C2214yescomplemented
36C3×#4023C2214yescomplemented
37#91573C2214yescomplemented
38#91570C2214yescomplemented
39#91566C2218yescomplemented
40#91543C2214yescomplemented
41#91542C2214yescomplemented
42#91468C2214yescomplemented
43#91451C2214yescomplemented
44#91420C2218yescomplemented
45#91419C2218yescomplemented
46#91418C2214yescomplemented
47#91417C2218yescomplemented
48#91416C2218yescomplemented
49Dic5×D12C2214yescomplemented
50#91405C2218yescomplemented
51#91350C2218yescomplemented
52#85735—51nomaximal, Hall
53C22×#711—32no—
54C5×#534—31no—
55#66138—32no—
56#65990—34no—
57#65982—34no—
58#65931—34no—
59#65930—34no—
60#65926—34no—
61#65920—34no—
62#65827—31no—
63C2×#4023D311yescharacteristic, complemented
64C24×C3×C5C2311yescharacteristic
65C23×D15C2312yes—
66C23×C5×D3C2312yes—
67C22×C3×C4×C5C2312yes—
68C2×#1220—28no—
69C2×#1220C2314yes—
70C2×#1220C2314yes—
71C22×Dic15C2312yes—
72C2×D60C2314yes—
73C2×C5×#67—28no—
74C2×C5×#67C2314yes—
75C2×C5×#67C2314yes—
76C22×C5×Dic3C2312yes—
77C2×C5×D12C2314yes—
78C22×C3×Dic5D418yescomplemented
79C22×C3×Dic5C2312yes—
80C22×C3×Dic5C2312yes—
81C22×C3×Dic5C2312yes—
82C22×C3×Dic5C2311yescharacteristic
83C22×C3×Dic5C2311yescharacteristic
84C2×#1204—28no—
85C2×#1204C2314yes—
86C2×#1204C2314yes—
87C2×#1203—28no—
88C2×#1203C2314yes—
89C2×#1203C2314yes—
90C2×#1203C2318yes—
91C2×#1200—28no—
92C2×#1200C2314yes—
93C2×#1200C2314yes—
94#6782—216no—
95#6782—216no—
96#6782—216no—
97#6782—216no—
98C2×D3×Dic5—28no—
99C2×D3×Dic5C2314yes—
100C2×D3×Dic5C2314yes—
101C3×C5×C22⋊C4C2314yes—
102#6714C2318yes—
103C4⋊Dic15C2314yes—
104C5×(C2×D3)⋊C4C2318yes—
105C5×C4⋊Dic3C2314yes—
106C3×C22⋊Dic5C2314yes—
107C3×#683C2318yes—
108C3×C4×Dic5C2314yes—
109C3×C4×Dic5C2314yes—
110#6668C2318yes—
111#6667C2318yes—
112(C2×D3)⋊Dic5C2318yes—
113C22×#235—52no—
114C2×C3×#117—51no—
115#5526—54no—
116#5525—54no—
117#5524—54no—
118#5524—54no—
119#5510—54no—
120#5493—52no—
121#5493—51no—
122#5493D511yescharacteristic, complemented
123#5477—54no—
124C22×C5×D4—32no—
125C22×C22⋊D5—32no—
126C22×C22⋊D5—32no—
127C23×Dic5—32no—
128C23×Dic5C2×D311yescharacteristic
129C2×#711—34no—
130C2×#711—34no—
131C2×#711—34no—
132C2×#711—34no—
133C2×#711—616no—
134C22×C4×D5—32no—
135C22×Dic10—32no—
136C5×#123—34no—
137C5×#121—34no—
138C5×#120—34no—
139C2×C5×C22⋊C4C2×D311yescharacteristic
140#4099—34no—
141C2×C22⋊Dic5C2×D311yescharacteristic
142#4086—34no—
143#4085—38no—
144#4085—38no—
145#4083—34no—
146#4081—34no—
147D4×Dic5—34no—
148#4076—34no—
149#4075—34no—
150C2×#683C2×D312yescomplemented
151C2×C4×Dic5C2×D312yescomplemented
152#4042—38no—
153#4039—34no—
154#4034—34no—
155#4032—34no—
156#4031—38no—
157#4030—38no—
158#4028—34no—
159#4027—38no—
160#4024—38no—
161#4023C2×D314yescomplemented
162#4023C2×D314yescomplemented
163#3449—151noSylow
164C23×C3×C5—42no—
165C23×C3×C5C2×D414yes—
166C23×C3×C5C2412yes—
167C23×C3×C5C2411yescharacteristic
168C22×D15—24no—
169C22×D15—44no—
170C22×D15C2414yes—
171C22×C5×D3—24no—
172C22×C5×D3—44no—
173C22×C5×D3C2414yes—
174C2×C3×C4×C5—42no—
175C2×C3×C4×C5C2414yes—
176#1220—28no—
177#1220—28no—
178C2×Dic15—24no—
179C2×Dic15C2414yes—
180D60—44no—
181C5×#67—28no—
182C5×#67—28no—
183C2×C5×Dic3—24no—
184C2×C5×Dic3C2414yes—
185C5×D12—44no—
186C2×C3×Dic5—42no—
187C2×C3×Dic5C2×D418yes—
188C2×C3×Dic5C2×D414yes—
189C2×C3×Dic5C2×D414yes—
190C2×C3×Dic5C2×D418yes—
191C2×C3×Dic5C2414yes—
192C2×C3×Dic5C2411yescharacteristic
193C2×C3×Dic5C2411yescharacteristic
194C2×C3×Dic5C2412yes—
195#1204—28no—
196#1204—28no—
197#1203—28no—
198#1203—28no—
199#1203—48no—
200#1200—28no—
201#1200—28no—
202D3×Dic5—28no—
203D3×Dic5—28no—
204C23×C3×C4—51no—
205C22×#67—52no—
206C22×#67C2×D512yes—
207C2×#235—54no—
208C2×#235—54no—
209C2×#235—54no—
210C2×#235—54no—
211C2×#235—1016no—
212C22×D12—52no—
213C22×C4×D3—52no—
214C22×C4×D3—52no—
215C22×Dic6—52no—
216C3×#117—54no—
217C3×#117—54no—
218C2×C3×C4⋊C4—52no—
219C2×C3×C22⋊C4—51no—
220C2×C3×C22⋊C4C2×D511yescharacteristic
221C2×C3×C42—52no—
222#930—54no—
223#930—58no—
224#930C2×D514yescomplemented
225#897—58no—
226#897—58no—
227#895—58no—
228#895—58no—
229#886—54no—
230#886—54no—
231#886—54no—
232#886C2×D514yescomplemented
233#882—54no—
234#882—54no—
235#882—54no—
236#882C2×D514yescomplemented
237#875—58no—
238#874—54no—
239C4×D12—54no—
240C4×D12—58no—
241C4×D12—54no—
242#869—54no—
243C24×C5—32no—
244C24×C5C22×D311yescharacteristic
245C23×D5—32no—
246C2×C5×D4—34no—
247C2×C5×D4—34no—
248C2×C5×D4—34no—
249C2×C5×D4—68no—
250C22×C4×C5—32no—
251C22×C4×C5C22×D312yes—
252C2×C22⋊D5—34no—
253C2×C22⋊D5—34no—
254C2×C22⋊D5—34no—
255C2×C22⋊D5—34no—
256C2×C22⋊D5—38no—
257C2×C22⋊D5—68no—
258C2×C22⋊D5—68no—
259C22×Dic5—32no—
260C22×Dic5—34no—
261C22×Dic5—34no—
262C22×Dic5—68no—
263C22×Dic5D1218yescomplemented
264C22×Dic5C22×D312yes—
265C22×Dic5C22×D312yes—
266C22×Dic5C22×D312yes—
267C22×Dic5C22×D311yescharacteristic
268C22×Dic5C22×D311yescharacteristic
269#711—616no—
270#711—616no—
271#711—616no—
272#711—616no—
273C2×D20—34no—
274C2×C4×D5—34no—
275C2×C4×D5—34no—
276C2×C4×D5—68no—
277C2×Dic10—34no—
278C2×Dic10—34no—
279C2×Dic10—68no—
280C5×C4⋊C4—34no—
281C5×C22⋊C4—38no—
282C5×C22⋊C4C22×D314yes—
283C22⋊Dic5—38no—
284C22⋊Dic5C22×D314yes—
285#685—38no—
286#685—38no—
287C4⋊Dic5—34no—
288#683—38no—
289#683C22×D318yes—
290C4×Dic5C22×D314yes—
291C4×Dic5C22×D314yes—
292C22×#41—152no—
293#537—154no—
294#536—154no—
295#535—154no—
296#534—31no—
297#534—152no—
298#534—151no—
299#532—154no—
300#525—154no—
301#525—154no—
302C2×#117—51no—
303C22×C3×C5—22no—
304C22×C3×C5—24no—
305C22×C3×C5—44no—
306C22×C3×C5C22×D414yes—
307C22×C3×C5C22×D412yes—
308C22×C3×C5C2511yescharacteristic
309C2×D15—24no—
310C2×D15—28no—
311C2×D15—48no—
312C2×C5×D3—24no—
313C2×C5×D3—28no—
314C2×C5×D3—48no—
315C3×C4×C5—24no—
316Dic15—24no—
317C3×Dic5—24no—
318C3×Dic5C22×D412yes—
319C3×Dic5C22×D414yes—
320C3×Dic5C22×D412yes—
321C5×Dic3—24no—
322C24×C3C22×D511yescharacteristic
323C23×D3—52no—
324C23×D3C22×D512yes—
325C22×C3×C4—58no—
326C22×C3×C4—51no—
327C22×C3×C4—51no—
328C22×C3×C4—52no—
329C22×C3×C4—52no—
330C22×C3×C4—52no—
331C22×C3×C4C22×D512yes—
332C2×#67—28no—
333C2×#67—54no—
334C2×#67—54no—
335C2×#67—108no—
336C2×#67C22×D514yes—
337C2×#67C22×D514yes—
338C22×Dic3—52no—
339C22×Dic3C22×D512yes—
340#235—1016no—
341#235—1016no—
342#235—1016no—
343#235—1016no—
344C2×D12—54no—
345C2×D12—54no—
346C2×D12—58no—
347C2×D12—54no—
348C2×D12—108no—
349C2×D12C22×D514yes—
350C2×C4×D3—54no—
351C2×C4×D3—54no—
352C2×C4×D3—54no—
353C2×C4×D3—54no—
354C2×C4×D3—108no—
355C2×C4×D3—108no—
356C2×Dic6—54no—
357C2×Dic6—54no—
358C2×Dic6—108no—
359C3×C4⋊C4—58no—
360C3×C22⋊C4—54no—
361C3×C22⋊C4C22×D514yes—
362C3×C42—54no—
363C3×C42—54no—
364(C2×D3)⋊C4—58no—
365(C2×D3)⋊C4—58no—
366(C2×D3)⋊C4—58no—
367(C2×D3)⋊C4C22×D518yes—
368C4⋊Dic3—54no—
369C4⋊Dic3—58no—
370C4⋊Dic3C22×D514yes—
371C23×C5—34no—
372C23×C5—42no—
373C23×C5—64no—
374C23×C5—124no—
375C23×C5C2×D1214yes—
376C23×C5C23×D312yes—
377C23×C5C23×D311yescharacteristic
378C22×D5—34no—
379C22×D5—64no—
380C22×D5—124no—
381C5×D4—68no—
382C5×D4—68no—
383C5×D4—124no—
384C2×C4×C5—34no—
385C2×C4×C5—42no—
386C2×C4×C5—64no—
387C2×C4×C5C23×D314yes—
388C22⋊D5—68no—
389C22⋊D5—68no—
390C22⋊D5—68no—
391C22⋊D5—68no—
392C22⋊D5—128no—
393C2×Dic5—34no—
394C2×Dic5—42no—
395C2×Dic5—64no—
396C2×Dic5—68no—
397C2×Dic5—68no—
398C2×Dic5C2×D1218yes—
399C2×Dic5C2×D1218yes—
400C2×Dic5C2×D1214yes—
401C2×Dic5C2×D1214yes—
402C2×Dic5C23×D314yes—
403C2×Dic5C23×D311yescharacteristic
404C2×Dic5C23×D311yescharacteristic
405C2×Dic5C23×D312yes—
406D20—124no—
407C4×D5—68no—
408C4×D5—68no—
409Dic10—68no—
410Dic10—68no—
411C2×#41—154no—
412C2×#41—154no—
413C2×#41—154no—
414C2×#41—154no—
415C2×#41—3016no—
416C22×Q8—152no—
417C22×D4—32no—
418C22×D4—152no—
419C22×D4—152no—
420C23×C4—51no—
421C23×C4—152no—
422C23×C4—152no—
423C4⋊Q8—154no—
424#127—154no—
425#124—158no—
426#123—34no—
427#123—154no—
428#123—154no—
429#123—154no—
430C22⋊Q8—158no—
431C22⋊Q8—158no—
432#121—34no—
433#121—158no—
434#121—158no—
435#121—158no—
436#121—154no—
437#120—34no—
438#120—154no—
439#120—154no—
440#120—154no—
441C4×D4—158no—
442C4×D4—154no—
443C4×D4—154no—
444#117—54no—
445#117—54no—
446C2×C4⋊C4—52no—
447C2×C22⋊C4—51no—
448C2×C22⋊C4D3×D511yescharacteristic
449C2×C42—52no—
450C2×C3×C5—24no—
451C2×C3×C5—22no—
452C2×C3×C5C23×D411yescharacteristic
453C2×C3×C5#58512yes—
454D15—44no—
455C5×D3—44no—
456C23×C3—42no—
457C23×C3D4×D514yes—
458C23×C3C23×D512yes—
459C23×C3C23×D511yescharacteristic
460C22×D3—24no—
461C22×D3—44no—
462C22×D3—54no—
463C22×D3—104no—
464C22×D3—204no—
465C22×D3C23×D514yes—
466C2×C3×C4—42no—
467C2×C3×C4—58no—
468C2×C3×C4—58no—
469C2×C3×C4—54no—
470C2×C3×C4—54no—
471C2×C3×C4—51no—
472C2×C3×C4—51no—
473C2×C3×C4—52no—
474C2×C3×C4—54no—
475C2×C3×C4—202no—
476C2×C3×C4C23×D514yes—
477#67—28no—
478#67—28no—
479#67—108no—
480#67—108no—
481C2×Dic3—24no—
482C2×Dic3—54no—
483C2×Dic3—104no—
484C2×Dic3C23×D514yes—
485D12—44no—
486D12—108no—
487D12—108no—
488D12—204no—
489D12—208no—
490C4×D3—108no—
491C4×D3—108no—
492C4×D3—108no—
493C4×D3—108no—
494Dic6—108no—
495Dic6—108no—
496C22×C5—22no—
497C22×C5—24no—
498C22×C5—44no—
499C22×C5—64no—
500C22×C5—68no—
501C22×C5—128no—
502C22×C5C22×D1214yes—
503C22×C5C22×D1212yes—
504C22×C5C24×D311yescharacteristic
505C2×D5—64no—
506C2×D5—68no—
507C2×D5—128no—
508C4×C5—24no—
509C4×C5—64no—
510Dic5—24no—
511Dic5—64no—
512Dic5C22×D1212yes—
513Dic5C22×D1214yes—
514Dic5C22×D1212yes—
515C24—32no—
516C24—152no—
517C24C2×D3×D511yescharacteristic
518#41—3016no—
519#41—3016no—
520#41—3016no—
521#41—3016no—
522C2×Q8—154no—
523C2×Q8—154no—
524C2×Q8—308no—
525C2×D4—34no—
526C2×D4—34no—
527C2×D4—34no—
528C2×D4—68no—
529C2×D4—154no—
530C2×D4—154no—
531C2×D4—158no—
532C2×D4—154no—
533C2×D4—154no—
534C2×D4—154no—
535C2×D4—308no—
536C2×D4—308no—
537C22×C4—32no—
538C22×C4—58no—
539C22×C4—51no—
540C22×C4—51no—
541C22×C4—52no—
542C22×C4—52no—
543C22×C4—52no—
544C22×C4—154no—
545C22×C4—154no—
546C22×C4—154no—
547C22×C4—154no—
548C22×C4—152no—
549C22×C4—308no—
550C22×C4—308no—
551C22×C4C2×D3×D512yes—
552C4⋊C4—34no—
553C4⋊C4—58no—
554C4⋊C4—158no—
555C4⋊C4—154no—
556C22⋊C4—38no—
557C22⋊C4—54no—
558C22⋊C4—158no—
559C22⋊C4—158no—
560C22⋊C4—158no—
561C22⋊C4C2×D3×D514yes—
562C42—54no—
563C42—54no—
564C3×C5#344911yescharacteristic, complemented, Hall
565C22×C3—22no—
566C22×C3—24no—
567C22×C3—44no—
568C22×C3C2×D4×D514yes—
569C22×C3C2×D4×D512yes—
570C22×C3C24×D511yescharacteristic
571C2×D3—24no—
572C2×D3—28no—
573C2×D3—48no—
574C2×D3—104no—
575C2×D3—108no—
576C2×D3—208no—
577C3×C4—24no—
578C3×C4—52no—
579C3×C4—54no—
580C3×C4—52no—
581C3×C4—104no—
582Dic3—24no—
583Dic3—104no—
584C2×C5—24no—
585C2×C5—22no—
586C2×C5—124no—
587C2×C5C23×D1211yescharacteristic
588C2×C5#596412yes—
589D5—124no—
590C23—34no—
591C23—42no—
592C23—64no—
593C23—124no—
594C23—154no—
595C23—304no—
596C23—604no—
597C23D5×D1214yes—
598C23C22×D3×D512yes—
599C23C22×D3×D511yescharacteristic
600Q8—308no—
601Q8—308no—
602D4—68no—
603D4—68no—
604D4—124no—
605D4—308no—
606D4—308no—
607D4—308no—
608D4—308no—
609D4—604no—
610D4—608no—
611C2×C4—34no—
612C2×C4—42no—
613C2×C4—58no—
614C2×C4—58no—
615C2×C4—54no—
616C2×C4—54no—
617C2×C4—51no—
618C2×C4—51no—
619C2×C4—52no—
620C2×C4—54no—
621C2×C4—64no—
622C2×C4—154no—
623C2×C4—202no—
624C2×C4—308no—
625C2×C4—308no—
626C2×C4—308no—
627C2×C4—308no—
628C2×C4—304no—
629C2×C4C22×D3×D514yes—
630C2×C3—24no—
631C2×C3—22no—
632C2×C3C22×D4×D511yescharacteristic
633C2×C3#6628512yes—
634D3—44no—
635D3—204no—
636C5#8573511yescharacteristic, complemented, Sylow
637C22—22no—
638C22—24no—
639C22—44no—
640C22—64no—
641C22—68no—
642C22—128no—
643C22—304no—
644C22—308no—
645C22—608no—
646C22C2×D5×D1214yes—
647C22C2×D5×D1212yes—
648C22C23×D3×D511yescharacteristic
649C4—24no—
650C4—52no—
651C4—54no—
652C4—52no—
653C4—64no—
654C4—104no—
655C4—304no—
656C3#1061819511yescharacteristic, complemented, Sylow
657C2—24no—
658C2—22no—
659C2—124no—
660C2—604no—
661C2C22×D5×D1211yescharacteristic
662C2#1175703412yes—
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC2×C3×C5×C22⋊C4C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#3449
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#6782C2semidirect
2. #11756252C2semidirect
3. C5×#5493C2semidirect
4. C2×C3×#4023C2semidirect
5. #11755428C2semidirect
6. #11754331C2semidirect
7. #11754330C2semidirect
8. #11752874C2semidirect
9. #11752873C2semidirect
10. #11752872C2semidirect
11. #11752871C2semidirect
16. C22×#1204C22semidirect
17. C22×#1203C22semidirect
18. C22×#1200C22semidirect
22. C2×#6782C22semidirect
23. C2×#6782C22semidirect
24. C22×D3×Dic5C22semidirect
26. #91819C22semidirect
27. #91775C22semidirect
28. #91771C22semidirect
29. C5×#930C22semidirect
30. C5×#886C22semidirect
31. C5×#882C22semidirect
33. C2×C3×#683C22semidirect
34. C2×C3×C4×Dic5C22semidirect
35. C3×#4023C22semidirect
36. C3×#4023C22semidirect
37. #91573C22semidirect
38. #91570C22semidirect
39. #91566C22semidirect
40. #91543C22semidirect
41. #91542C22semidirect
42. #91468C22semidirect
43. #91451C22semidirect
44. #91420C22semidirect
45. #91419C22semidirect
46. #91418C22semidirect
47. #91417C22semidirect
48. #91416C22semidirect
49. Dic5×D12C22semidirect
50. #91405C22semidirect
51. #91350C22semidirect
63. C2×#4023D3semidirect
78. C22×C3×Dic5D4semidirect
122. #5493D5semidirect
150. C2×#683C2×D3semidirect
151. C2×C4×Dic5C2×D3semidirect
161. #4023C2×D3semidirect
162. #4023C2×D3semidirect
224. #930C2×D5semidirect
232. #886C2×D5semidirect
236. #882C2×D5semidirect
263. C22×Dic5D12semidirect
564. C3×C5#3449semidirect
636. C5#85735semidirect
656. C3#10618195semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#67822
#117562522
C5×#54932
C2×C3×#40232
#117554282
#117543312
#117543302
#117528742
#117528732
#117528722
#117528712
#106181953
#857355